Magnetic field of rotating wire I was doing some questions on the magnetic field when a charged body is rotated on its general axis, but I had a doubt that what if it's not the general axis but somewhere else. Like, for example, take a ring and take an axis along the diameter of the ring and to find its magnetic field somewhere at a point in front of it. Now is this approach right that we think that it's rotating in a way that it would seem like a spherical shell and then solve for it?
 A: I think that perhaps the best way to solve the general problem of finding the magnetic field of an arbitrary charge distribution rotating over an arbitrary axis is to simply integrate the magnetic field produced by the differential charge elements that make up the charged object, which is:
$$\mathbf{d \vec{B}} =\frac{μ_0}{4π}  \frac{ dq \cdot \mathbf{\vec{V}} \times \mathbf{\vec{R}}}{r^3}$$
You would calculate the velocity of each differential element on the object relative to the point's reference frame and the displacement vector from the point to each bit, then apply the formula and integrate over the object, i.e. $\iiint\mathbf{d \vec{B}}$. There are many schemes by which you could integrate this, and spherical shells would be one of them (though almost certainly not the best one).
I suspect that it would be time-varying for many such objects, and possibly even non-analytic (thus numerical computer integration would be required). If you took your example of the ring rotated by an axis along the diameter that was perpendicular to the plane of the ring, there are some places where the ring is sometimes very near and sometimes very far, resulting in a complicated time-varying magnetic field, that you're better of just using a computer to solve.
Sorry, but I'm pretty sure there is no nice analytic way to get magnetic fields of off-axis rotating charged bodies (though I suspect someone has invented a perturbative theory for small displacements).
